Adjacent-numbered pages:
BP154 BP155 BP156 BP157 BP158  *  BP160 BP161 BP162 BP163 BP164

convoluted, unwordable

Harry E. Foundalis

All examples are black and white images, partitioned by lines such that crossing a line switches the background color and the foreground color. (Sometimes it is not clear which is "background" and which is "foreground".) In the space between two dividing lines, there is a black and white scene; the outlines of the shapes are curves dividing black from white. Images sorted left are such that each outline-curve present in a scene that comes in contact non-tangentially with a dividing line continues across the dividing line, across which the black and white sides of it switch.

Examples (especially right) usually have ambiguity to some degree; depending on how a person reads the images, dividing lines may be confused for curves within a scene.

Adjacent-numbered pages:
BP519 BP520 BP521 BP522 BP523  *  BP525 BP526 BP527 BP528 BP529

fuzzy, unwordable, anticomputer, traditional, blackwhiteinvariant

Aaron David Fairbanks

Examples on the left are also known as "Dyck words".

https://en.wikipedia.org/wiki/Dyck_language

Adjacent-numbered pages:
BP951 BP952 BP953 BP954 BP955  *  BP957 BP958 BP959 BP960 BP961

easy, nice, precise, allsorted, unwordable, notso, sequence, traditional, inductivedefinition, preciseworld, left-listable, right-listable

Aaron David Fairbanks

On the left, each row and column could be labeled by a certain object or concept; on the right this is not so.

More specifically: on the left, each row and each column is associated with a certain object or concept; there is a rule for combining rows and columns to give images; it would be possible without changing the rule to extend with new rows/columns or delete/reorder any existing columns. On the right, this is not so; the rule might be about how the images must relate to their neighbors, for example.

All examples show grids of squares with an image in each square, such that there is some "rule" the images within the grid obey.

Left examples are a generalized version of the analogy grids seen in BP361. Any analogy a : b :: c : d shown in a 2x2 grid will fit on the left here.

To word the solution with mathematical jargon, "defines a (simply described) map from the Cartesian product of two sets vs. not so." Another equivalent solution is "columns (alternatively, rows) illustrate a consistent set of one-input operations." It is always possible to imagine the columns as inputs and the rows as operations and vice versa.

There is a trivial way in which any example can be interpreted so that it fits on the left side: imagine each row is assigned the list of all the squares in that row and each column is assigned its number, counting from the left. But each grid has a clear rule that is simpler than this.

BP1258 is a similar idea: "any square removed could be reconstructed vs. not." Examples included left here usually fit left there, but some do not e.g. EX9998.

See BP979 for use of similar structures but with one square removed from the grid.

Adjacent-numbered pages:
BP976 BP977 BP978 BP979 BP980  *  BP982 BP983 BP984 BP985 BP986

nice, convoluted, unwordable, notso, teach, structure, rules, grid, miniworlds

grid_of_images_with_rule [smaller | same | bigger]
zoom in left (grid_of_analogies)

Aaron David Fairbanks

Right:

All are "all but one are ___"; all but one are black.

All are "every other is ___"; every other is solid polygons.

All are "gradually becoming ___"; gradually becoming thickly outlined.

Left:

All but one are "all but one are ___".

Every other is "every other is ___".

Gradually becoming "gradually becoming ___".

Here is another way of putting it:

Call it "meta" when the whole imitates its parts, and call it "doubly-meta" when the whole imitates its parts with respect to the way it imitates its parts. Left are doubly-meta, while right are just meta.

Here is a more belabored way of putting it:

Call something like "is star-shaped" a "rule". An object can satisfy a rule.

Call something like "all but one are ___" a "rule-parametrized rule". A collection of objects can satisfy a rule-parametrized rule with respect to a particular rule.

On the right: every collection fits the same rule-parametrized rule (with respect to various rules); furthermore the collection of collections fits that same rule-parametrized rule (with respect to some unrelated rule that collections can satisfy).

On the left: The collection of collections fits a rule-parametrized rule with respect to the rule of fitting that rule-parametrized rule (with respect to various rules).

Previously, an unintended solution to this BP was "not all groups share some noticeable property vs. all do." It is hard to come up with examples foiling this alternative solution because the rule-parametrized rule (see explanation above) usually has to do with not all objects in the collection fitting the rule. (See BP568, which is about BP ideas that are always overridden by a simpler solution.) The example EX10108 "all five are 'all five are ___'" was added, foiling the alternative solution.

The right side of this Problem is a subset of BP999^left.

Adjacent-numbered pages:
BP993 BP994 BP995 BP996 BP997  *  BP999 BP1000 BP1001 BP1002 BP1003

"Odd one out with respect to what property is the odd one out" would not fit left: even though this example does seem doubly-meta, it is not doubly-meta in the right way. There is no odd one out with respect to the property of having an odd one out.

Similarly, consider "gradual transition with respect to what the gradual transition is between", etc. Instead of having the form "X 'X __' ", this is more like "X [the __ appearing in 'X __']". Examples like these two could make for a different Bongard Problem.

hard, unwordable, challenge, overriddensolution, infodense, contributepairs, funny, rules, miniworlds

zoom in right

Aaron David Fairbanks

The description in terms of rectangles was noted by Sridhar Ramesh when he solved this.

All examples in this Bongard Problem feature arced line segments connected at endpoints; these segments do not cross across one another and they are nowhere vertical; they never double back over themselves in the horizontal direction.

Furthermore, in each example, there is a single leftmost point and a single rightmost point, and every segment is part of a path bridging between them. So, there is a topmost total path of segments and bottommost total chain of segments.

Any picture on the left can be turned into a subdivided rectangle by the process of expanding points into vertical lines.

Here is another answer:

"Right examples: some junction point has a single line coming out from either the left or right side."

If there is some junction point with only a single line coming out from a particular side, the point cannot be expanded into a vertical segment with two horizontal segments bookending its top and bottom (as it would be if this were a subdivision of a rectangle).

And this was the original, more convoluted idea of the author:

"Start with a string along the top path. Sweep it down, region-by-region, until it lies along the bottom path. The string may only enter a region when it fully covers that region's top edge and likewise it must exit by fully covering the bottom edge. Only in left images can this process be done so that no segment of the string ever hesitates."

Quite convoluted when spelled out in detail, but not terribly complicated to imagine visually. (See the keyword unwordable.)

The string-sweeping answer is the same as the rectangle answer because a rectangle represents the animation of a string throughout an interval of time. (A horizontal cross-section of the rectangle represents the string, and the vertical position is time.) Distorting the rectangle into a new shape is the same as animating a string sweeping across that new shape.

In particular, shrinking vertical lines of a rectangle into points means just those points of the string stay still as the string sweeps down.

The principle that horizontal lines subdividing the original rectangle become the segments in the final picture corresponds to the idea that the string must enter or exit a single region all at once.

BP1129 started as an incorrect solution for this Bongard Problem. Anything fitting right in BP1130 fits right in BP1129.

Adjacent-numbered pages:
BP1125 BP1126 BP1127 BP1128 BP1129  *  BP1131 BP1132 BP1133 BP1134 BP1135

hard, unwordable, solved

[smaller | same | bigger]

Aaron David Fairbanks

Left-sorted examples are sometimes called autobiographical or self-descriptive numbers.

https://oeis.org/A349595

https://en.wikipedia.org/wiki/Self-descriptive_number

See BP1147 for a similar idea.

BP1149 was inspired by this.

Adjacent-numbered pages:
BP1143 BP1144 BP1145 BP1146 BP1147  *  BP1149 BP1150 BP1151 BP1152 BP1153

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

Leo Crabbe

Inspired by BP1148.

Adjacent-numbered pages:
BP1144 BP1145 BP1146 BP1147 BP1148  *  BP1150 BP1151 BP1152 BP1153 BP1154

nice, precise, unwordable, notso, handed, leftright, left-narrow, sequence, preciseworld, left-listable, right-listable

Aaron David Fairbanks

Adjacent-numbered pages:
BP1150 BP1151 BP1152 BP1153 BP1154  *  BP1156 BP1157 BP1158 BP1159 BP1160

abstract, unwordable, creativeexamples, right-unknowable, traditional, finishedexamples, rules

Leo Crabbe

Operations depicted in right-sorted examples are called "commutative".

"Order matters" here means that if the objects in the top half were to switch places, the output would look different.

https://en.wikipedia.org/wiki/Commutative_property

Adjacent-numbered pages:
BP1152 BP1153 BP1154 BP1155 BP1156  *  BP1158 BP1159 BP1160 BP1161 BP1162

nice, abstract, unwordable, notso, structure, rules, miniworlds

Leo Crabbe